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## Appendix 2: PWBMsim’s Attribute Transition Functions

1) The incidence of fertility ($f$) for females aged 14-49 is governed by the fertility rate values specific to the potential mother’s age, ethnicity, and education.

$$f_t=f(\text{age}_t,\text{ethnicity},\text{education})$$

2) Mortality incidence in PWBM is governed by the mortality rates calculated by the Social Security Administration. However, those rates are available by ethnicity and gender and further decomposed by race, education, and marital status using data from the National Center for Health Statistics. The probability of dying in year t is conditional on the person’s age in year t, gender, and ethnicity, educational attainment, and marital status in year $t$: $$d_t=d(\text{age},\text{gender},\text{ethnicity},\text{education},\text{marital status})$$

3) The number of immigrants (the foreign born) that enter PWBMsim’s population ($P$) in any year is given by the above immigration rates ($i$) for legal and undocumented immigrants. $$I_t=i_t xP$$

4) The evolution of an individuals’ legal status (native-born/naturalized/legal immigrant/unauthorized immigrant) is governed by:

$$l_t=d(\text{age},\text{gender},\text{ethnicity},\text{education},\text{marital status})$$

5) Labor force status transitions (not-working/wage-only/self-employment only/wage and self-employment) for each PWBMsim individual are governed by the person’s demographic status and the previous labor force status. The relationship can be described by the function. The function F(.) below consists of a set of empirically determined probabilities. That is, the elements of this table control transition across employment status states:

$$LFS_{t}=F(LFS_{t-1} \mid \text{ethnicity},\text{gender},\text{age},\text{education})$$

6) PWBMsim’seducational attainment transitions are calibrated to generate the ratios and ratio-trends in education prevalence rates calculated from micro-data, by ethnicity and gender. The education transition control parameters govern the likelihood that a person of age, g, in period t, will acquire ($\delta{e}$) an additional year of education in that period: $$P(\delta{e})=E(e_{t-1}\mid \text{age}_t,\text{ethnicity},\text{gender})$$

7) Marriage ($m$) and divorce ($d$) transitions are also governed by marriage and divorce transition functions:

\begin{align} m_t &= M(m\mid \text{ethnicity},\text{gender},\text{age},\text{education}\ldots) \\ d_t &= D(d\mid \text{ethnicity},\text{gender},\text{age},\text{education}\ldots) \end{align}

8) Total capital assignments to each PWBMsim individual in period $t (k_{it})$ are governed by capital holdings estimated from private, public, and foreign sourced capital obtained from various micro-data surveys and government reports. The assignments for future years assume that per-capita values by age and gender will grow with the population of the previous year’s simulated productivity growth.

$$K_t=K(k_{it} \mid \text{Ethnicity},\text{Age},\text{Gender},\text{Education})$$

9) Disability status (transitions into and out of disability status) is governed by the probability of becoming disabled in year t, given the PWBMsim individuals age, gender, and race.

$$\delta_{t}=d(\delta_{t-1} \mid \text{age},\text{gender},\text{ethnicity})$$

10) PWBM individual ($log$) labor earnings of individual $i$ in period $t$, ($Z_{it}$) are conditioned on all of the person’s attributes including race, age, gender, marital status, family size, disability status, labor force status, education, and interactions of these attributes with each other: Only full- and part-time workers have positive earnings. The controlling regression includes a first order auto-regressive effect of past earnings. The parameters $\theta$ captures the influence of individual attributes and the parameter $\alpha$ captures the influence of the available capital stock that workers use in the economy.

$$ln⁡Z_{it} =f(X_{it};\Theta)=ln⁡(1-\alpha)+(1-\alpha)\sum_{j=1}^{k} \Theta_{j} X_{jit} + u_{it}$$